Question

determining key features of spring motion
a weight hangs from a spring and bobs 3 inches above and below an equilibrium position.
the weight takes 8 seconds to complete a full cycle (period = 8).
when graphed, time is on the horizontal axis and position relative to equilibrium is on the vertical axis.
which statements are true? choose three correct answers.
the graph is compressed horizontally by a factor of $\frac{pi}{4}$
the frequency is 8.
the amplitude is 6.
the graph is stretched vertically by a factor of 3.
the frequency is $\frac{1}{8}$
this question requires at least 3 answers.

Explanation:

Step1: Identify amplitude

Amplitude = max displacement from equilibrium = 3 inches.

Step2: Calculate frequency

Frequency $f = \frac{1}{\text{Period}} = \frac{1}{8}$

Step3: Analyze vertical stretch

Vertical stretch factor = amplitude = 3.

Step4: Analyze horizontal compression

Standard period of $\sin(t)$ is $2\pi$. Horizontal scale factor $k = \frac{2\pi}{\text{Period}} = \frac{2\pi}{8} = \frac{\pi}{4}$. This means horizontal compression by $\frac{\pi}{4}$.

Step5: Evaluate each statement

1. "The graph is compressed horizontally by a factor of $\frac{\pi}{4}$": True (matches Step4)
2. "The frequency is 8": False (frequency is $\frac{1}{8}$)
3. "The amplitude is 6": False (amplitude is 3)
4. "The graph is stretched vertically by a factor of 3": True (matches Step3)
5. "The frequency is $\frac{1}{8}$": True (matches Step2)

Answer:

A. The graph is compressed horizontally by a factor of $\frac{\pi}{4}$
D. The graph is stretched vertically by a factor of 3
E. The frequency is $\frac{1}{8}$